Editing Chirality (mathematics) (section)

==Chirality in two dimensions==
[[File:Bracelets33.svg|thumb|300px|The colored [[Necklace (combinatorics)|necklace]] in the middle is '''chiral''' in two dimensions; the two others are '''achiral'''.<br>This means that as physical necklaces on a table the left and right ones can be rotated into their mirror image while remaining on the table. The one in the middle, however, would have to be picked up and turned in three dimensions.]]
[[File:Triangle.Scalene.svg|thumb|250px|A [[scalene triangle]] does not have mirror symmetries, and hence is a [[chiral polytope]] in 2 dimensions.]]
In two dimensions, every figure which possesses an [[axis of symmetry]] is achiral, and it can be shown that every ''bounded'' achiral figure must have an axis of symmetry. (An ''axis of symmetry'' of a figure <math>F</math> is a line <math>L</math>, such that <math>F</math> is invariant under the mapping <math>(x,y)\mapsto(x,-y)</math>, when <math>L</math> is chosen to be the <math>x</math>-axis of the coordinate system.) For that reason, a [[triangle]] is achiral if it is [[equilateral triangle|equilateral]] or [[isosceles triangle|isosceles]], and is chiral if it is [[Triangle#By_lengths_of_sides|scalene]].

Consider the following pattern:
:[[File:Krok 6.svg|320px]]

This figure is chiral, as it is not identical to its mirror image:
:[[File:Krok 6 mirrored.png|320px]]

But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a [[frieze group]] generated by a single [[glide reflection]].